Patil et al. [1]} estimate the probability of collision under robot state uncertainty by truncating the state distributions. In [2]}, future state distributions are predicted and the uncertainties are used to compute bounded collision probabilities. Lee et al. [3]} use sigma hullsSigma hulls are convex hulls of the geometry of
individual robot links transformed according to the sigma
points in joint space [3]}. to formulate collision avoidance constraints in terms of the signed distance to the obstacles. Du Toit and Burdick [5]}, Park et al. [6]} compute the collision probability by marginalizing the joint distribution between the robot and obstacle location. The distributions are assumed to be Gaussian and the marginalization is computed with an indicator function that is true under the collision condition. However, since there is no closed-form solution to this formulation, an approximation is assumed. Furthermore, Park et al. compute an upper bound for the collision probability. An approximation is computed using Monte Carlo Integration in [7]}, albeit computationally intensive. Another impressive work that uses Monte Carlo approach is Monte Carlo Motion
Planning (MCMP) [8]}. This approach first solves a deterministic motion planning problem with inflated obstacles and then adjusts the inflation to compute a path that is exactly as safe as desired.
